With those words, fearing King Saul, David expressed his deep desire to flee his troubles. To fly as does a dove would be much faster than even the swiftest of feet. Those wishing-for-wings words follow sentences of confession.
What thoughts trouble you? What keeps you up at night? Does anxiety drain your energy? Does the fear of death halt your faith-based courage? David answered questions like those with soul-bearing honesty. So should we. Once we do, we need to take the next step.
What step? The step that David took.
While David remained flightless, he did not remain hopeless. In his hour (actually hours, days, months, etc.) of trouble, David chose to call on God.
In your hour of trouble, call on God.
When you stare at the ceiling, stop the whole counting sheep thing and talk to God (quietly if your spouse is next to you).
If you're nervous, ask God to calm your spirit.
Once we learn to do those things, we can experience the peace about which the psalmist wrote.
You and I will not grow wings. But we will feel like soaring eagles when we call on God.